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Definite integrals by substitution

Problem 4.606 · medium

Evaluate \( \displaystyle \int_{0}^{2} 4 x \sqrt{x^{2} + 4}\, dx \).

Let u = x**2 + 4; then du = 2*x dx, which is in the integrand up to a constant.
Reviewed
\[ \left. x^{2} + 4 \right|_{\substack{ x=0 }} = 4 \]
The lower limit in u.✓ Proved
\[ \left. x^{2} + 4 \right|_{\substack{ x=2 }} = 8 \]
The upper limit in u.✓ Proved
\[ \int\limits_{4}^{8} 2 \sqrt{u}\, du = - \frac{32}{3} + \frac{64 \sqrt{2}}{3} \]
Integrate in u between the new limits; no back-substitution needed.✓ Proved

Answer \( - \frac{32}{3} + \frac{64 \sqrt{2}}{3} \)

✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	numerical quadrature of the original integral

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly applies u-substitution, properly transforms the limits of integration, and accounts for the constant factor in the differential. The final result matches the stated answer.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies u-substitution, properly transforms the limits of integration, and accounts for the constant factor in the differential. The final result matches the stated answer.
gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly applies u-substitution, adjusts the limits of integration, and computes the definite integral accurately.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/definite_substitution, checked 2026-10-07 with SymPy 1.14.0.